18 CANONICAL DECOMPOSITIONS OF A MEASURE 209 \fd\fi\ =1 JnfM- Since /t = h • LU| with |/i(an)| = 1 for all n (13.16.3), we w= 1 oo have /x(<jD{fln}) = pn with |/?J = yn, and f/rfju = £ Ai/(*n) for every /z-integ- J n= 1 rable function/. Conversely, consider an arbitrary denumerable sequence (an) of points of X, and a sequence (yn) of real numbers >0 such that £ yn < + 00 for all aneK compact subsets K of X. We have already seen (13.1.3) that /^v(/) = f ynf(an) «= i is a positive measure on X. With the same notation as above, it is clear that v = sup((pDn • v), and since <pDn - v is concentrated on Dn and hence on D, it n follows that v is concentrated on D. Also we have v({an}) = yn. For if K is a compact neighborhood of an, then for each e > 0 there exists an integer m such that ]T yp <^ e. If /: X -» [0, 1] is a continuous mapping which takes «peK,p£m the value 1 at an, and the value 0 on X — K and at points ak such that k ^ n and k <a m (4.5.2), then we have v({an}) g v(/) <; yn H- s; on the other hand we may choose the neighborhood K such that v(K) ^ v({an}) + e (13.7.9), and a fortiori yn ^ v(/) ^ v({an}) + e. Since s was arbitrary, our assertion is proved. PROBLEMS 1. Let p, a be two atomic measures on a locally compact space X, and let M, N be the smallest sets carrying |p|, |cr , respectively. Show that p, cr are disjoint if and only if M r\ N = 0. Hence give an example of an atomic measure v on the interval I = [0, 1] of R such that I is the support (13.19) of v+ and of v". 2. (a) Let ^ be a positive measure on a locally compact space X. If A c X is universally measurable and not ja-negligible, show that there exists a positive measure v carried by A such that v ^ 0 and v <i p,. (Observe that A contains a compact set K which is not /x-negligible.) (b) Let M be a universally measurable subset of X. For a positive measure /x to be carried by M it is necessary and sufficient that v should be disjoint from every positive measure carried by X — M (use (a)). (c) If M is closed in X, show that the vector subspace of M(X) consisting of the measures carried by M is vaguely closed in M(X). (d) Show that Lebesgue measure on 1= [0,1] is the vague limit of a sequence of atomic measures carried by a fixed denumerable subset D of I. The subspace of M(I) consisting of measures carried by D is therefore not vaguely closed.le set fr)l f°r ^1 ° e fri* ^i], then we have